Parameter Statistic and Random Samples A parameter is

Parameter, Statistic and Random Samples • A parameter is a number that describes the population. It is a fixed number, but in practice we do not know its value. • A statistic is a function of the sample data, i. e. , it is a quantity whose value can be calculated from the sample data. It is a random variable with a distribution function. Statistics are used to make inference about unknown population parameters. • The random variables X 1, X 2, …, Xn are said to form a (simple) random sample of size n if the Xi’s are independent random variables and each Xi has the sample probability distribution. We say that the Xi’s are iid. week 1 1

Example – Sample Mean and Variance • Suppose X 1, X 2, …, Xn is a random sample of size n from a population with mean μ and variance σ2. • The sample mean is defined as • The sample variance is defined as week 1 2

Goals of Statistics • Estimate unknown parameters μ and σ2. • Measure errors of these estimates. • Test whether sample gives evidence that parameters are (or are not) equal to a certain value. week 1 3

Sampling Distribution of a Statistic • The sampling distribution of a statistic is the distribution of values taken by the statistic in all possible samples of the same size from the same population. • The distribution function of a statistic is NOT the same as the distribution of the original population that generated the original sample. • The form of theoretical sampling distribution of a statistic will depend upon the distribution of the observable random variables in the sample. week 1 4

Sampling from Normal population • Often we assume the random sample X 1, X 2, …Xn is from a normal population with unknown mean μ and variance σ2. • Suppose we are interested in estimating μ and testing whether it is equal to a certain value. For this we need to know the probability distribution of the estimator of μ. week 1 5

Claim • Suppose X 1, X 2, …Xn are i. i. d normal random variables with unknown mean μ and variance σ2 then • Proof: week 1 6

Recall - The Chi Square distribution • If Z ~ N(0, 1) then, X = Z 2 has a Chi-Square distribution with parameter 1, i. e. , • Can proof this using change of variable theorem for univariate random variables. • The moment generating function of X is • If , all independent then • Proof… week 1 7

Claim • Suppose X 1, X 2, …Xn are i. i. d normal random variables with mean μ and variance σ2. Then, are independent standard normal variables, where i = 1, 2, …, n and • Proof: … week 1 8

t distribution • Suppose Z ~ N(0, 1) independent of X ~ χ2(n). Then, • Proof: week 1 9

Claim • Suppose X 1, X 2, …Xn are i. i. d normal random variables with mean μ and variance σ2. Then, • Proof: week 1 10

F distribution • Suppose X ~ χ2(n) independent of Y ~ χ2(m). Then, week 1 11

Properties of the F distribution • The F-distribution is a right skewed distribution. • i. e. • Can use Table 7 on page 796 to find percentile of the F- distribution. • Example… week 1 12

The Central Limit Theorem • Let X 1, X 2, …be a sequence of i. i. d random variables with E(Xi) = μ < ∞ and Var(Xi) = σ2 < ∞. Let Then, for - ∞ < x < ∞ where Z is a standard normal random variable and Ф(z)is the cdf for the standard normal distribution. • This is equivalent to saying that Z ~ N(0, 1). converges in distribution to • Also, i. e. converges in distribution to Z ~ N(0, 1). week 1 13

Example • Suppose X 1, X 2, …are i. i. d random variables and each has the Poisson(3) distribution. So E(Xi) = V(Xi) = 3. • The CLT says that as n ∞. week 1 14

Examples • A very common application of the CLT is the Normal approximation to the Binomial distribution. • Suppose X 1, X 2, …are i. i. d random variables and each has the Bernoulli(p) distribution. So E(Xi) = p and V(Xi) = p(1 - p). • The CLT says that as n ∞. • Let Yn = X 1 + … + Xn then Yn has a Binomial(n, p) distribution. So for large n, • Suppose we flip a biased coin 1000 times and the probability of heads on any one toss is 0. 6. Find the probability of getting at least 550 heads. • Suppose we toss a coin 100 times and observed 60 heads. Is the coin fair? week 1 15